Vremenske i svjetlosne plohe u Lorentz-Minkowskijevom prostoru

master's thesis

• Title (english): Time and light surfaces in Lorentz-Minkowski space
• Author: Filip Martinović
• Mentor: Željka Milin Šipuš (mentor)
• Committee member: Željka Milin Šipuš (predsjednik povjerenstva)
• Committee member: Dražen Adamović (član povjerenstva)
• Committee member: Marko Vrdoljak (član povjerenstva)
• Committee member: Marko Erceg (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2021-07-22, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Cilj ovog rada jest obraditi primjere vremenskih i svjetlosnih ploha Lorentz-Minkowskijevog prostora \(\mathbb{R}^3_1\). Da bismo proučili te primjere, potrebno je razraditi diferencijalnu geometriju trodimenzionalnog realnog prostora s nedegeneriranom metrikom indeksa 1. U prvom poglavlju ispitujemo kakvu geometriju inducira ta metrika na sam prostor \(\mathbb{R}^3_1\), klasifciramo vektore i potprostore tog prostora te ispitujemo svojstva ortogonalnosti. Zatim, u drugom poglavlju ispitujemo svojstva krivulja tog prostora, defniramo njihovu Freneteovu bazu, izvodimo Frenet-Serreteove formule te dokazujemo analogone fundamentalnog teorema krivulja. U trećom poglavlju proučavamo plohe prostora \(\mathbb{R}^3_1\), klasifciramo ih na prostorne, vremenske i svjetlosne te proučavamo kakvu geometriju na njima inducira metrika prostora \(\mathbb{R}^3_1\). Konačno, koristeći razrađenu teoriju, ispitujemo primjere vremenskih i svjetlosnih ploha, dakle ploha čija je inducirana metrika nedegenrirana indefnitna odnosno degenerirana i time nalazimo primjere ploha koje nemaju analogon u klasičnoj euklidskoj geometriji.

Abstract (english)

Goal of this paper is to examine examples of timelike and lightlike surfaces in Lorentz- Minkowski space \(\mathbb{R}^3_1\). To do so, we need to study differential geometry of the threedimensional real space with nondegenerate metric of index 1. In first chapter we inspect geometric properties that this metric induces on space \(\mathbb{R}^3_1\), we classify vectors and subspaces of this space and we investigate properties of orthogonality. Next, in second chapter we explore properties of curves in this space, we define corresponding Frenet frame, we derive Frenet-Serret equations and we prove theorems analogous to standard fundamental theorems for curves. In third chapter we study surfaces of \(\mathbb{R}^3_1\), we classify them as spacelike, timelike and lightlike and we examine what kind of geometry does the metric of \(\mathbb{R}^3_1\) induces on the surfaces. Lastly, by using developed theory we inspect examples of timelike and lightlike surfaces, that is surfaces with induced nondegenerate indefinite metric or degenerate metric and with that we find examples of surfaces that have no analogues in standard Euclidean geometry.

• Keywords: diferencijalna geometrija trodimenzionalnog realnog prostora; Freneteova baza; Frenet-Serreteove formule
• Keywords (english): differential geometry of the threedimensional real space; Frenet frame; Frenet-Serret equations
• Language: croatian
• URN:NBN: urn:nbn:hr:217:350949
• Study programme: Title: Pure Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2021-09-16 09:22:05